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Vanishing theorems for pseudo-effective line bundles

Xiankui Meng, Chenghao Qing, Xiangyu Zhou

Abstract

In the present paper, we establish a general Kawamata-Viehweg-Kollár-Nadel type vanishing theorem for higher direct images in terms of numerical dimension for closed positive currents on compact Kähler manifolds, unifying a number of important vanishing theorems.

Vanishing theorems for pseudo-effective line bundles

Abstract

In the present paper, we establish a general Kawamata-Viehweg-Kollár-Nadel type vanishing theorem for higher direct images in terms of numerical dimension for closed positive currents on compact Kähler manifolds, unifying a number of important vanishing theorems.
Paper Structure (9 sections, 22 theorems, 96 equations)

This paper contains 9 sections, 22 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L^2$-spaces and $L^2$-estimates
  4. Multiplier ideal sheaves and Numerical dimension
  5. Regularization of closed positive currents
  6. Higher direct images of pseudo-effective line bundles
  7. Girbau type theorem
  8. Kawamata-Viehweg type Theorem
  9. Applications

Key Result

Theorem 1.1

The upper regularized multiplier ideal sheaf is exactly the multiplier ideal sheaf, that is, $\mathcal{I}_+(\varphi)=\mathcal{I}(\varphi)$ if $\varphi$ is a quasi-plurisubharmonic function

Theorems & Definitions (31)

  • Theorem 1.1: strong openness GZ15-a
  • Theorem 1.2: Kaw82Vie82DemSmall
  • Theorem 1.3: Cao14GZ15-a
  • Theorem 1.4: Kol86-a
  • Theorem 1.5: Fuj18FM21
  • Theorem 1.6: Mat16Ohs84
  • Theorem 1.7: QZ25
  • Theorem 1.8: =Theorem \ref{["MainThm1'"]}
  • Theorem 1.9: =Theorem \ref{["MainThm2'"]}
  • Theorem 1.10: =Theorem \ref{["MainThm3'"]}
  • ...and 21 more